House Minority Leader Nancy Pelosi of Calif. gestures as she speaks to reporters during a news conference on Capitol Hill in Washington on Wednesday.

WASHINGTON — Frustrated Democrats lamented Wednesday that persistent problems with new health care exchanges have inflicted damage on the public’s perception of the already unpopular “Obamacare” — with some lawmakers insisting President Barack Obama should ensure those responsible lose their jobs.

Emerging from a closed-door briefing with health officials from the Obama administration, House Democrats appeared to have at least as many questions as answers about how and when the beleaguered website will be fixed. Although they resolved not to let setbacks with one aspect of the health law outshine the parts that are working, they griped that the shoddy website had given Republicans an opening to do just that.

“I think the president needs to man up, find out who was responsible and fire them,” Rep. Richard Nolan, D-Minn., said after the briefing. He said Obama should tell Democrats when the problems will be fixed so they can prepare to move on. “You don’t get many second chances to get a good first impression.”

Nolan wasn’t the only one.

“Somebody should be held accountable,” said Rep. Xavier Becerra, D-Calif. “Absolutely.”

The briefing with House Democrats came as the Obama administration was appealing to its allies in Congress, on Wall Street and across the country to stick with the health care law despite embarrassing problems that continue to crop up. On Wednesday, lawmakers heard from Gary Cohen and Julie Bataille, two high-level officials with the Centers for Medicare and Medicaid Services, a federal agency with major responsibility for the website where millions of Americans are expected to purchase insurance.

Democrats say they requested the briefing. A spokesman for House Speaker John Boehner, R-Ohio, said House Republicans were arranging to hold a similar briefing with health officials in the coming days.

Echoing Democratic leaders and even Obama himself, Democrats said it was unacceptable that the website’s debut had been so flawed. But Rep. Rosa DeLauro, D-Conn., said it was critical Democrats not lose sight of the bigger picture and law’s other benefits.

Obama has turned to longtime adviser Jeffrey Zients, a veteran management consultant, to provide advice to help fix the system. And Obama has said he’s instituted a “tech surge,” bringing in leading technology talent to repair the painfully slow and often unresponsive website. But the administration has repeatedly declined to say how long that will take, raising questions about whether the full extent of the problems has been fully determined.

“They were reluctant to give a date — I don’t blame them — on the fixes,” said Rep. Janice Schakowsky, D-Ill. But they said the problems would be fixed in time for people to get enrolled by Jan. 1, the day that coverage through the exchanges begins.

The website’s troubled debut was overshadowed by the partial government shutdown that started the same day the website went live. Last week, Obama and Democrats walked away from a no-holds-barred fight with Republicans over debt and spending with a remarkable degree of unity, made all the more prominent by the deep GOP divisions the standoff revealed.

The debt-and-spending crisis averted for now, the spotlight has shifted to Obama’s health care law and the web-based exchanges, beset by malfunctions, where Americans are supposed to be able to shop for insurance. The intensified focus has increased the pressure on Democrats to distance themselves from Obama’s handling of the website’s rollout as both parties demand to know what went wrong and why.

As the administration races to fix the website, it’s deploying the president and top officials to urge his supporters not to give up.

“By now you have probably heard that the website has not worked as smoothly as it was supposed to,” Obama said Tuesday in a video message recorded for Organizing for America, a nonprofit group whose mission is to support Obama’s agenda. “But we’ve got people working overtime in a tech surge to boost capacity and address the problems. And we are going to get it fixed.”

The group has been organizing a multitude of events and social media campaigns around the health care law’s implementation. OFA said those efforts will continue, but the group is not adjusting its strategy in response to the website’s issues.

Meanwhile, Vice President Joe Biden and top White House officials held a call with business leaders Tuesday about the health law and other issues. Business Forward, a trade group friendly to the White House, said the administration asked the group to invite leaders to hear directly from Biden.